Volume 1 / Part 5 — *212. THE SERIES OF SEGMENTS.
Reading edition
*212. THE SERIES OF SEGMENTS.
Summary of *212.
The series of segments or sections of a series may be ordered by the relation of inclusion, after the manner considered in *210. Since, as was shown in *211, sections and segments have the properties assigned to in the hypothesis of *210, the resulting series are such that every class has either a maximum or a sequent, and either a minimum or a precedent; i.e. the series of segments or sections are Dedekindian. Most of the properties of the series of sections and of the series of segments which have no maximum, only require that the original relation should be connected. The properties of the series of segments in general require also that the original relation should be transitive.
We denote the series of segments by , putting
We then have, in virtue of *210·13 and *211·61,
*212·23.
In like manner, for the series of segments which have no maximum, we put and we have
*212·22.
We do not need a special notation for the series of sections, since, in virtue of *211·13, it is or . Thus, by *212·23,
*212·24.
We begin the number with various propositions on the fields, etc. of these relations, and on the conditions for their existence. We have
*212·132.
*212·133.
*212·14.
*212·152.
*212·17.
*212·172.
Of the next set of propositions (*212·2—·25), several have already been mentioned. An important proposition is
*212·25.
for this shows that the series of segments contains a series similar to .
We take up next the application of the propositions of *210 to the series of sections and segments. We show that if , and are series (*212·3), and that if is also transitive, is a series (*212·31). We have
*212·322.
*212·34.
so that every class of sections has both an upper limit or maximum and a lower limit or minimum (*212·35).
We then prove similar propositions for and , except that in place of *212·34 we have
*212·431.
*212·53.
The reason of the difference from *212·34 is that the product of an existent class of segments may not be a segment. Suppose, for example, the segments are all those that contain a given term , where has no immediate successor; then their logical product is , which is a section but not a segment.
We have next (*212·6—·667) a number of propositions on the limits and maxima of sub-classes of in the series . The interest of this subject lies in its relation to irrationals. If is a class contained in and having no limit or maximum, is contained in , and has a limit in . We may call this limit an irrational segment. There is no irrational term in , because in there is no limit to ; but the limit, in , of may be called irrational, because it corresponds to no term in . It should be observed that (as will be proved in Section F) if is similar to the series of rationals, is similar to the series of real numbers.
The most useful propositions in this subject are:
*212·6.
*212·601.
*212·602.
*212·61.
*212·632.
*212·661.
This shows that every limit in the series of segments is a limit of a class of what we may call rational segments (i.e. segments of the form ), namely it is the limit of .
*212·667.
This shows that the segments (other than ) which are limits of classes of segments are the segments (other than ) which have no maximum in .
The number ends with a set of propositions (*212·7—·72) on the relations of the sections and segments of two correlated series. If is a correlator of with , then (with its converse domain limited) is a correlator of with , with and with (*212·71·711·712). Hence
*212·72.
This proposition is used in the next number, and also in *271.
*212·01.
*212·02.
*212·1.
*212·11.
Dem.
*212·12.
Thus has the same connection with as has with . When is transitive, also has the same connection with as has with . The following proposition makes these facts more explicit.
*212·121.
Dem.
*212·122.
*212·123.
*212·13.
*212·131.
Dem.
*212·132.
Dem.
*212·133.
Dem.
*212·134.
*212·14.
Dem.
*212·141.
Dem.
*212·142.
Dem.
*212·15.
*212·151.
The converse implication does not hold in this case. For the existence of , it is necessary that should contain classes having no maximum.
*212·152.
*212·153.
Dem.
*212·154.
Dem.
*212·155.
*212·156.
Dem.
*212·16.
Dem.
*212·161.
Dem.
*212·162.
Dem.
*212·17.
Dem.
*212·171.
*212·172.
*212·173.
*212·18.
Dem.
*212·181.
The above proposition is used in *252·43.
*212·2.
*212·21.
*212·22.
*212·23.
*212·24.
*212·25.
Dem.
The following propositions, down to *212·55, consist of applications of the propositions of *210, where the of that number is replaced by , , or , and the is replaced by , i.e. by , , or . The propositions which follow are important, since the use of segments, especially in connection with continuity, depends largely upon them.
*212·3.
*212·31.
*212·32.
We write , instead of putting below the line, because, when we have to deal with an expression not consisting of a single letter, it is inconvenient to write it as a suffix, especially when it contains a suffix itself, as in this case.
*212·321.
*212·322.
*212·33.
Dem.
*212·331.
Dem.
*212·34.
*212·35.
*212·36.
Dem.
*212·4.
*212·401.
*212·402.
*212·41.
*212·411.
*212·42.
The cases considered in *212·411 and *212·42 are not mutually exclusive, since if , we have .
*212·421.
Dem.
*212·43.
Thus with regard to the lower end of a class chosen out of , we have three cases to distinguish: (1) if , is the minimum; (2) if , is the lower limit; (3) if , is the lower limit.
*212·431.
Dem.
*212·44.
*212·45.
Dem.
The proofs of the following propositions are exactly analogous to those of the corresponding propositions on .
*212·5.
*212·501.
*212·502.
*212·51.
*212·511.
*212·52.
This proposition includes *212·511, since, if , we have
*212·53.
The proof proceeds as in *212·431.
*212·54.
*212·55.
The following propositions are concerned with the relations of maxima, limits and sequents in and respectively. The series , which is ordinally similar to , is contained in ; and if has a maximum or limit in , the maximum or limit of in is or . In this way, a series (namely ) which has the same ordinal properties as can be placed in a certain Dedekindian series (namely ) in such a way that the classes which have limits in are those whose correlates have limits which are members of , while those whose correlates have limits which are not members of are those which have neither a maximum nor a limit in . These relations are important in many connections. For example, if is of the type of the rationals, is of the type of the real numbers: corresponds to the irrationals, and classes contained in but having a limit not belonging to correspond to series of rationals having an irrational limit. In the original series , there are no irrational limits; but if is a class in and having no limit, has an irrational limit in .
*212·6.
Dem.
*212·601.
*212·602.
Dem.
*212·61.
*212·62.
Dem.
*212·621.
Dem.
*212·63.
*212·631.
Dem.
*212·632.
Dem.
*212·633.
Dem.
*212·65.
Dem.
*212·651.
Dem.
*212·652.
Dem.
*212·653.
Dem.
*212·66.
Dem.
*212·661.
Dem.
*212·662.
*212·663.
Dem.
*212·664.
Dem.
*212·665.
Dem.
*212·666.
Dem.
*212·667.
*212·7.
Dem.
*212·701.
*212·702.
*212·71.
Dem.
*212·711.
*212·712.
*212·72.